mobius-topology: A Toolkit for the Möbius Magnetic Topology (MMT) Series

mobius-topology is an installable Python toolkit that makes the Möbius Magnetic Topology (MMT) series reproducible in about ten minutes: it computes Gauss linking numbers of thin-tube field lines directly from the explicit parameterization, reproduces all seven theorems of MMT-II (Paper 9), and exposes the Unified Replacement Theorem as one-line formulas. The package has four modules: - linking: thin-tube helix and Möbius field-line parameterization (Lemma U: a Möbius field line is pointwise identical to the helix of slope s+1/2), the Gauss linking integral (midpoint rule, blockwise accumulation), writhe, twist and self-linking; - replacement: the complete linking-number formula system — co-oriented -(q/p)k^2 (Thm 1), counter-oriented same-radius 0 (Thm 2), counter-oriented distinct-radii +(q/p)k^2 (Thm 3), their Möbius counterparts under the slope replacement q/p -> q/p+1/2 (Thms 4-5), the four-line net-linking conservation 0 (Thm 6), and the thin-tube twist/writhe analytics of the Wr+Tw=SL decomposition (Thm 7); - convergence: the two-parameter sampling threshold N·rho·|dtheta| ≳ C(k) of MMT-II §A.2, with a warning when k exceeds the calibrated range (k ≤ 14); - visualization: linking-vs-k and embedding-angle plots. Two example notebooks are included. 01_basic_usage reproduces Theorem 1, Lemma U, Theorem 7 and the convergence scan in about five minutes. 02_magnetic_island demonstrates the first observable signature of the series (MMT-II §14): the tube-radius-order sign -> E-M interlinking sign -> magnetic-island chirality (synthetic demo of constructed thin-tube field lines; the interface convention for real diagnostics is defined by the MMT-III paper). Two short Chinese tutorials explain how to compute the Gauss linking number of Möbius field lines in 50 lines of NumPy, and why the Möbius-to-torus passage is a one-line replacement. A smoke test reproduces all seven theorems. Numerical verification: all seven theorems of MMT-II are reproduced with errors